I will give a survey of my recent results in this area: 1) with Jozef Bobok we have studied the dynamical properties of a typical continuous map of [0,1] which preserve the Lebesgue measure, 2) with Eleonora Catisigeras we have studied the structure of invariant measures of a typical continuous map of closed compact manifold (with or without boundary), and also for a typical homeomorphism.